If the given matrix is ______ matrix, the assignment problem is called balanced problem - Mathematics and Statistics

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If the given matrix is ______ matrix, the assignment problem is called balanced problem

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Solution

Square

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Chapter 2.7: Assignment Problem and Sequencing - Q.2

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RELATED QUESTIONS

A job production unit has four jobs A, B, C, D which can be manufactured on each of the four machines P, Q, R and S. The processing cost of each job is given in the following table:

         Jobs

 

 

                          Machines

P

Q

R

S

                Processing Cost (Rs.)

 

A

31

25

33

29

B

25

24

23

21

C

19

21

23

24

D

38

36

34

40

 How should the jobs be assigned to the four machines so that the total processing cost is minimum?


Solve the following minimal assignment problem and hence find the minimum value : 

  I II III IV
A 2 10 9 7
B 13 2 12 2
C 3 4 6 1
D 4 15 4 9

 


Solve the following minimal assignment problem : 

Machines A B C D E
M1 27 18 20 21
M2 31 24 21 12 17
M3 20 17 20 16
M4 21 28 20 16 27

Solve the following minimal assignment problem and hence find minimum time where  '- ' indicates that job cannot be assigned to the machine : 

Machines Processing time in hours
A B C D E
M1 9 11 15 10 11
M2 12 9 - 10 9
M3 - 11 14 11 7
M4 14 8 12 7 8

Solve the following maximal assignment problem :

Branch Manager Monthly Business ( Rs. lakh)
A B C D
P 11 11 9 9
Q 13 16 11 10
R 12 17 13 8
S 16 14 16 12

 


A departmental head has three jobs and four subordinates. The subordinates differ in their capabilities and the jobs differ in their work
contents. With the help of the performance matrix given below, find out which of the four subordinates should be assigned which jobs ?

Subordinates Jobs
I II III
A 7 3 5
B 2 7 4
C 6 5 3
D 3 4 7

In a factory there are six jobs to be performed each of which should go through two machines A and B in the order A - B. The processing timing (in hours) for the jobs arc given here. You are required to determine the sequence for performing the jobs that would minimize the total elapsed time T. What is the value of T? Also find the idle time for machines · A and B.

Jobs J1 J2 J3 J4 J5 J6
Machine A 1 3 8 5 6 3
MAchine B 5 6 3 2 2 10

A job production unit has four jobs A, B, C, D which can be manufactured on each of the four machines P, Q, R and S. The processing cost of each job for each machine is given in the following table:

Jobs Machines
(Processing Cost in ₹)
P Q R S
A 31 25 33 29
B 25 24 23 21
C 19 21 23 24
D 38 36 34 40

Find the optimal assignment to minimize the total processing cost.


Five wagons are available at stations 1, 2, 3, 4, and 5. These are required at 5 stations I, II, III, IV, and V. The mileage between various stations are given in the table below. How should the wagons be transported so as to minimize the mileage covered?

  I II III IV V
1 10 5 9 18 11
2 13 9 6 12 14
3 3 2 4 4 5
4 18 9 12 17 15
5 11 6 14 19 10

The assignment problem is said to be unbalance if ______


The assignment problem is said to be balanced if ______.


Choose the correct alternative :

The assignment problem is said to be balanced if it is a ______.


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In an assignment problem if number of rows is greater than number of columns then


The objective of an assignment problem is to assign ______.


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When an assignment problem has more than one solution, then it is _______ optimal solution.


Fill in the blank :

An _______ is a special type of linear programming problem.


In an assignment problem, if number of column is greater than number of rows, then a dummy column is added.


State whether the following is True or False :

It is not necessary to express an assignment problem into n x n matrix.


Solve the following problem :

A dairy plant has five milk tankers, I, II, III, IV and V. These milk tankers are to be used on five delivery routes A, B, C, D and E. The distances (in kms) between the dairy plant and the delivery routes are given in the following distance matrix.

  I II III IV V
A 150 120 175 180 200
B 125 110 120 150 165
C 130 100 145 160 175
D 40 40 70 70 100
E 45 25 60 70 95

How should the milk tankers be assigned to the chilling center so as to minimize the distance travelled?


Choose the correct alternative:

The assignment problem is generally defined as a problem of ______


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Assignment Problem is special case of ______


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When an assignment problem has more than one solution, then it is ______


Choose the correct alternative:

The assignment problem is said to be balanced if ______


In an assignment problem if number of rows is greater than number of columns, then dummy ______ is added


State whether the following statement is True or False:

The objective of an assignment problem is to assign number of jobs to equal number of persons at maximum cost


State whether the following statement is True or False:

In assignment problem, if number of columns is greater than number of rows, then a dummy row is added


State whether the following statement is True or False: 

In assignment problem each worker or machine is assigned only one job


Three jobs A, B and C one to be assigned to three machines U, V and W. The processing cost for each job machine combination is shown in the matrix given below. Determine the allocation that minimizes the overall processing cost.

    Machine
    U V W
Jobs A 17 25 31
B 10 25 16
C 12 14 11

(cost is in ₹ per unit)


A departmental head has four subordinates and four tasks to be performed. The subordinates differ in efficiency and the tasks differ in their intrinsic difficulty. His estimates of the time each man would take to perform each task is given below:

    Tasks
    1 2 3 4
Subordinates P 8 26 17 11
  Q 13 28 4 26
  R 38 19 18 15
  S 9 26 24 10

How should the tasks be allocated to subordinates so as to minimize the total manhours?


Find the optimal solution for the assignment problem with the following cost matrix.

    Area
    1 2 3 4
  P 11 17 8 16
Salesman Q 9 7 12 6
  R 13 16 15 12
  S 14 10 12 11

Assign four trucks 1, 2, 3 and 4 to vacant spaces A, B, C, D, E and F so that distance travelled is minimized. The matrix below shows the distance.

  1 2 3 4
A 4 7 3 7
B 8 2 5 5
C 4 9 6 9
D 7 5 4 8
E 6 3 5 4
F 6 8 7 3

Choose the correct alternative:

Number of basic allocation in any row or column in an assignment problem can be


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North – West Corner refers to ______


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If number of sources is not equal to number of destinations, the assignment problem is called ______


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A car hire company has one car at each of five depots a, b, c, d and e. A customer in each of the fine towers A, B, C, D and E requires a car. The distance (in miles) between the depots (origins) and the towers(destinations) where the customers are given in the following distance matrix.

  a b c d e
A 160 130 175 190 200
B 135 120 130 160 175
C 140 110 155 170 185
D 50 50 80 80 110
E 55 35 70 80 105

How should the cars be assigned to the customers so as to minimize the distance travelled?


A natural truck-rental service has a surplus of one truck in each of the cities 1, 2, 3, 4, 5 and 6 and a deficit of one truck in each of the cities 7, 8, 9, 10, 11 and 12. The distance(in kilometers) between the cities with a surplus and the cities with a deficit are displayed below:

    To
    7 8 9 10 11 12
From 1 31 62 29 42 15 41
2 12 19 39 55 71 40
3 17 29 50 41 22 22
4 35 40 38 42 27 33
5 19 30 29 16 20 33
6 72 30 30 50 41 20

How should the truck be dispersed so as to minimize the total distance travelled?


A dairy plant has five milk tankers, I, II, III, IV and V. Three milk tankers are to be used on five delivery routes A, B, C, D and E. The distances (in kms) between the dairy plant and the delivery routes are given in the following distance matrix.

  I II III IV V
A 150 120 175 180 200
B 125 110 120 150 165
C 130 100 145 160 170
D 40 40 70 70 100
E 45 25 60 70 95

How should the milk tankers be assigned to the chilling center so as to minimize the distance travelled?


A job production unit has four jobs P, Q, R, and S which can be manufactured on each of the four machines I, II, III, and IV. The processing cost of each job for each machine is given in the following table:

Job Machines
(Processing cost in ₹)
I II III IV
P 31 25 33 29
Q 25 24 23 21
R 19 21 23 24
S 38 36 34 40

Find the optimal assignment to minimize the total processing cost.


A department store has four workers to pack goods. The times (in minutes) required for each worker to complete the packings per item sold is given below. How should the manager of the store assign the jobs to the workers, so as to minimize the total time of packing?

Workers Packing of
  Books Toys Crockery Cutlery
A 3 11 10 8
B 13 2 12 12
C 3 4 6 1
D 4 15 4 9

A job production unit has four jobs P, Q, R, S which can be manufactured on each of the four machines I, II, III and IV. The processing cost of each job for each machine is given in the following table :

Job Machines
(Processing cost in ₹)
I II III IV
P 31 25 33 29
Q 25 24 23 21
R 19 21 23 24
S 38 36 34 40

Complete the following activity to find the optimal assignment to minimize the total processing cost.

Solution:

Step 1: Subtract the smallest element in each row from every element of it. New assignment matrix is obtained as follows :

Job Machines
(Processing cost in ₹)
I II III IV
P 6 0 8 4
Q 4 3 2 0
R 0 2 4 5
S 4 2 0 6

Step 2: Subtract the smallest element in each column from every element of it. New assignment matrix is obtained as above, because each column in it contains one zero.

Step 3: Draw minimum number of vertical and horizontal lines to cover all zeros:

Job Machines
(Processing cost in ₹)
I II III IV
P 6 0 8 4
Q 4 3 2 0
R 0 2 4 5
S 4 2 0 6

Step 4: From step 3, as the minimum number of straight lines required to cover all zeros in the assignment matrix equals the number of rows/columns. Optimal solution has reached.

Examine the rows one by one starting with the first row with exactly one zero is found. Mark the zero by enclosing it in (`square`), indicating assignment of the job. Cross all the zeros in the same column. This step is shown in the following table :

Job Machines
(Processing cost in ₹)
I II III IV
P 6 0 8 4
Q 4 3 2 0
R 0 2 4 5
S 4 2 0 6

Step 5: It is observed that all the zeros are assigned and each row and each column contains exactly one assignment. Hence, the optimal (minimum) assignment schedule is :

Job Machine Min.cost
P II `square`
Q `square` 21
R I `square`
S III 34

Hence, total (minimum) processing cost = 25 + 21 + 19 + 34 = ₹`square`


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